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PHYS 3900 Homework Set #02

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PHYS 3900 Homework Set #02
Physics 3900
Spring 2016
University of Georgia
Instructor: HBSchüttler
PHYS 3900 Homework Set #02
Part 1 = HWP 2.01, 2.02, 2.03. Due: Wed. Jan. 27, 2016, 4:00pm
Part 2 = HWP 2.04, 2.05, 2.06. Due: Wed. Feb. 3, 2016, 4:00pm
All textbook problems assigned, unless otherwise stated, are from the textbook by M. Boas
Mathematical Methods in the Physical Sciences, 3rd ed. Textbook sections are identified
as ”Ch.cc.ss” for textbook Chapter ”cc”, Section ”ss”. Complete all HWPs assigned: only
two of them will be graded; and you don’t know which ones! Read all ”Hints” before you
proceed! Make use of the ”PHYS3900 Homework Toolbox”, posted on the course web site.
Do not use the calculator, unless so instructed! All arithmetic, to the extent required, is
either elementary or given in the problem statement. State all your answers in terms of
√
real-valued elementary functions (+, −, /, ×, power, root, , exp, ln, sin, cos, tan, cot,
arcsin, arccos, arctan, arcot, ...) of integer numbers, e and π; in terms of i where needed;
and in terms of specificpinput variables, as stated in each problem. So, for example, if the
result is, say, ln(7/2) + (e5 π/3), or 179/2 , or (16−4π)−10 , then just state that as your final
answer: no need to evaluate it as decimal number by calculator! Simplify final results to the
largest extent possible; e.g., reduce fractions of integers to the smallest denominator etc..
In some problems below, you will need to apply (without proof) d’Alembert’s Ratio Test
(RT) which can be used to test convergence of a series of real or complex numbers
Sn :=
n
X
Tj ,
with n = 0, 1, 2, ... ,
j=0
constructed from a sequence of real or complex numbers Tj with j = 0, 1, 2, .... The RT is
applicable only if (1) an integer jo ≥ 0 exists such that Tj 6= 0 for all integer j ≥ jo ; and (2)
the limit
ρ := lim |Tj+1 /Tj |
j→∞,j≥jo
either exists with a finite limit ρ ≥ 0; or if it is is infinite (meaning: the sequence ρj :=
|Tj+1 /Tj | grows beyond all lower bounds for j → ∞). The RT states:
If ρ < 1 the series Sn is convergent, i.e., it does have a limit,
S :=
∞
X
Tj ;
j=0
and it is, in fact, absolutely convergent. If, on the other hand, ρ > 1, the series Sn is
divergent. If ρ = 1 the series Sn may be either convergent or divergent, i.e., the RT does not
tell you anything about convergence. However, for power series, the condition ρ = 1 defines
the boundary of the convergence disk: it can be used to find both the radius, Rc (a.k.a. the
radius of convergence), and the center, zc , of the convergence disk.
1
Physics 3900
Spring 2016
University of Georgia
Instructor: HBSchüttler
HWP 02.01: Evaluate the following expressions, without any calculator, either in cartesian
(CC) or polar (PC) coordinates, as stated. For CC, with z = x + iy, give x and y (with real
x and y). For PC, with z = reiθ , give r and θ (with real r ≥ 0 and real θ). For PC, any θ
with
0 ≤ θ ≤ 2π ,
but only those (Watch Out!), are acceptable. You may use, without proof, any tool in the
Toolbox and any known property of real elementary functions. In parts (c) and (d), variables
a, b, c and d denote real numbers.
(2 − i)99997 √
√
√
(a) (1 − 3i)5 ( 2 + i 8)7 (CC); (b) √
(CC);
99995
( 3 + i 2)
(c) (d)
(7c − 7ia)3 (a 6= 0 or c 6= 0) (CC);
49i |(|a| + i|c|)3 |
∗
1
e−i(b−2i ln(2)) − ei(d+2i ln(2)) e−2iπ/3+id + e−i(b−7π/3) , for cos(b + d) =
5
√
√
(e) i6 − 5 + 2/i + i3 − 3i − 7 (PC).
(PC);
HWP 02.02: This problem extends the definition of the conventional, real square root
√
function, ..., to arbitrary complex numbers z = x + iy with x = Re z and y = Im z.
(a) Show that, for every complex z = x + iy, there exists exactly one complex number
w = u + iv, with u = Re w and v = Im w, obeying the following two conditions:
w2 = z
(2.2.1)
and
either Re w > 0 or [Re w = 0 and Im w ≥ 0]
(2.2.2)
Express both u and v as functions of x and y.
This uniquely defined complex number w = u + iv is then referred to as the general complex
square root of z, and denoted by
√
z := w .
c
Hints: Write out Eq.(2.2.1) as two real equations, expressed in terms of the real variables
u, v, x and y. Eliminate v to solve for u; then solve for v. Prove that such a solution (u, v)
always exists, for any choice of (x, y), while carefully distinguishing the two cases u 6= 0 and
u = 0. Prove that condition (2.2.2) makes the choice of this solution (u, v) unique, i.e., there
is only one solution of Eq.(2.2.1) which obeyes (2.2.2).
(b) Use the results from Part (a) to calculate
√
√
√
√
√
−4 , c 4i , c −4i , c +1 + i ,
c +4 ,
c
√
−1 − i ,
c
√
c
+1 − i ,
√
c
−1 + i .
Sketch each result as a point, properly labeled, in the complex plane, using your calculator
to evaluate any real square roots you may need.
2
Physics 3900
Spring 2016
University of Georgia
Instructor: HBSchüttler
(c) Use the result from Part (a) to prove that for purely real numbers, z=x with y=0,
p
√
√
√
z
=
x
if
x
≥
0
;
z
=
i
|x|
if x < 0 ;
c
c
where
√
... is the familiar, conventional square-root function defined for positive real numbers.
(d) Consider any complex number z = x + iy written in polar coordinates,
z = |z| eiθ ,
with θ chosen so that −π < θ ≤ π. Use the result from Part (a) to show that
p
√
z =
|z| eiθ/2 .
c
(e) Use the result from Part (d) to show for any non-zero complex z = x + iy 6= 0 that
p
p
√
√
(1/z) = 1/c z
if x ≥ 0 or y 6= 0 ;
(1/z) = −1/c z
if x < 0 and y = 0 .
c
c
HWP 02.03: Recall from your intro physics course that the electrical power dissipated in
a circuit, i.e., the rate at which electrical energy is being consumed by the circuit, is the
product of the voltage applied and the curent driven through the circuit by that voltage.
This is a very general result, based only on the energy conservation law, and therefore holds
even for time-dependent voltages and currents. (For an example, see the circuit shown in
Fig. 02.04 below.)
Therefore, if an alternating voltage Ṽ (t) := Vo cos(ωt − φV ) drives an alternating current
˜ := Io cos(ωt − φI ) through an electrical circuit, the electrical power dissipated in the
I(t)
circuit, i.e. the circuit’s rate of energy consumption at time t, is
˜ .
P̃ (t) = Ṽ (t)I(t)
For practical purposes, knowing the exact t-dependence of this (rapidly) oscillating power
function is often ”too much information”: one is actually not interested in knowing the
”instantaneous power” consumption P̃ (t) at each time t, but rather only its long-time average
over many periods oscillation, defined by:
Z
1 to +τ
P = lim
P̃ (t)dt
τ →∞ τ t
o
R t +τ
Note here, too dtP̃ (t) is the amount of energy consumed in the circuit over a time interval
[to , to + τ ], of duration τ , starting at some time to .
(a) Show that P is independent of to and that it can be written in terms of the complex
voltage and current amplitudes V := Vo e−iφV and I := Io e−iφI as
1
P = Re(V ∗ I) .
2
While there are other stategies to get this result, in order to practice your ”complex calculus”
skills, you should solve this problem specifically in the following steps:
3
Physics 3900
Spring 2016
University of Georgia
Instructor: HBSchüttler
(1) Express the real voltage and current as real parts of the complex voltage and current,
˜ = Re(Ieiωt ). Use the identity Re(X) = (X + X ∗ )/2 (for any
by Ṽ (t) = Re(V eiωt ) and I(t)
complex number X) to re-write the foregoing real parts in terms of V eiωt and Ieiωt and their
respective complex conjugates, V ∗ e−iωt and I ∗ e−iωt , using (eiωt )∗ = e−iωt . (2) Then take
the product of Ṽ and I˜ to express and expand out P̃ (t) in terms of four products of V eiωt
and Ieiωt and their respective complex conjugates, using also the identities eiωt e−iωt = 1 and
e±iωt e±iωt = e±2iωt . (3) For each of the resulting (four) V I-product terms in P̃ (t), evaluate
the definite t-integrals (from to to to + τ ), by proving and then using the fact that for any
complex z 6= 0, F (t) := (iz)−1 eizt is an indefinite integral (anti-derivative) of f (t) := eizt ,
i.e., dF (t)/dt = f (t). You can prove this last equation, for example, by using the MacLaurin
expansion of eizt in powers of t. (4) I trust you remember from your intro calculus course how
to get from an indefinite to a definite integral! (5) Add up all the (four) definite integrals
thus obtained, divide by τ and take the limit τ → ∞. Make use of the fact that |eiθ | = 1 for
any real θ, hence |eiθ(τ ) |/τ → 0 for τ → ∞ and any real-valued function θ(τ ). (6) Show that
V ∗ I + V I ∗ = 2Re(V ∗ I) to complete the derivation.
(b) Use the result from (a) and the definition of V and I, that is, V := Vo e−iφV and
I := Io e−iφI , to show that P can also be written as:
1
P = Vo Io cos(φ)
2
where φ is the phase angle between current and voltage:
φ := φI − φV .
Footnote: Recall here that the t-dependent complex voltage V eiωt and t-dependent complex
current Ieiωt represent rotating vectors in the complex plane, with the tips of the vectors
moving along circles of radii |V | ≡ Vo and |I| = Io , respectively, both of them rotating at
angular velocity ω. The phase angle φ is then the constant angle enclosed between these
two rotating vectors. Visualize this! It’s an important way of thinking about sinusoidally
oscillating quantities, that is useful far beyond just AC circuit analysis.
(c) If the circuit is ”linear”, i.e., if the current ampltitude is proportional to the voltage
˜ are related by some complex-valued
amplitude, then sinusoidally oscillating Ṽ (t) and I(t)
impedance function Z which is determined by the design (”innner workings”) of the circuit
and generally dependent on angular oscillation frequency ω. That is, the corresponding
complex amplitudes V and I then obey the generalized Ohm’s Law:
V = ZI .
Let Z be written in polar coordinates as Z ≡ |Z|eiθZ . Show that the result from (a) or (b)
can then be written as
1
1 Vo2
cos(φ)
P = |Z|Io2 cos(φ) =
2
2 |Z|
with
φ = θZ .
4
Physics 3900
Spring 2016
University of Georgia
Instructor: HBSchüttler
Fig. 02.04 - c CtJ
HWP 02.04: [Note: Before doing this problem, you may find it helpful to review the
application of Kirchhoff rules to derive the equivalent resistance R for three ohmic resistors
R1 , R2 and R3 connected in parallel to a battery in a simple DC circuit. The problem below,
especially Part (a), is a generalization of these Kirchhoff rule ideas to an AC circuit.]
˜
An alternating current I(t)
:= Io cos(ωt − φI ) is driven by an applied voltage Ṽ (t) :=
Vo cos(ωt − φV ) through the following parallel RLC-circuit, containing a resistor R with
impedance ZR = R, an inductance L with impedance ZL = (iωL), and a capacitor C with
impedance ZC = 1/(iωC), as shown in Fig. 02.04.
(a) Show that the complex voltage and current amplitudes, V := Vo e−iφV and I := Io e−iφI ,
are related by a generalized Ohm’s Law,
V = ZI ,
with an effective impedance Z (for the RLC-circuit as whole), given by:
1
Z
=
1
1
1
+
+
ZR ZL ZC
=
5
1
1 + i ωC −
.
R
ωL
Physics 3900
Spring 2016
University of Georgia
Instructor: HBSchüttler
Hints: Assume (without proof) the Kirchhoff junction rule stating for the case of this
˜ = I˜R (t) + I˜L (t) + I˜C (t) where I˜R (t) ≡ Re(IR eiωt ), I˜L (t) ≡ Re(IL eiωt ), and
circuit that I(t)
I˜C (t) ≡ Re(IC eiωt ) are the alternating currents flowing into R, L and C, respectively, as
shown in Fig. 02.04, with complex amplitudes IR , IL , and IC , respectively. Also assume
(without proof) the Kirchhoff loop rule, stating for the case of this circuit that each of the
three voltage drops, ṼR (t), ṼL (t), and ṼC (t), across the three circuit elements, R, L and C,
respectively, is the same and equal to the applied voltage Ṽ (t).
Re-state these Kirchhoff rule results in terms of the complex amplitudes I, IR , IL , IC , V ,
˜ I˜R , I˜L , I˜C ) and
VR , VL , VC , associated with the corresponding real oscillating currents (I,
voltages (Ṽ , ṼR , ṼL , ṼC ).
Then use (without proof) the fact that each of the three circuits elements R, L and C obeys
”it’s own” generalized Ohm’s Law, namely, in terms of complex amplitudes:
IR =
1
VR ,
ZR
IL =
1
VL ,
ZL
IC =
1
VC
ZC
Use all the foregoing to first express IR , IC and IL in terms of V and impedances ZR , ZL
and ZC ; and then I in terms of V and impedances ZR , ZL and ZC . Then simply use the
definition Z := V /I to get the result for 1/Z stated above.
(b) Use the results from Part (a) and from HWP 02.03 Part (c) to show that the timeaveraged power dissipation P in the circuit, expressed as a function of Io , R, L, C and ω,
can be written as:
RIo2
1
P =
2 1 + R2 [ωC − 1/(ωL)]2
Sketch a very rough graph of P (ω) as a function of ω for fixed Io , R, L and C. Indicate on
the graph the asymptotic behavior i.e., the approximate power laws P (ω) ∼ constant × ω p
for ω → 0 and for ω → ∞.
Hint: Let Y := 1/Z = |Y |e−iθZ . Find Re(Y ) and Im(Y ) and |Y |. Draw Y as a vector
in the complex plane to prove/see that cos(φ) = Re(Y )/|Y | with φ = θZ . Also note that
1/|Z| = |Y |. Use Y instead of Z throughout the calculation.
(c) Find the ”resonance frequency” ωo , expressd in terms of R, L and C, where P (ω) reaches
its maximum (at fixed Io ).
Hint: Since P (ω) > 0, it reaches its maximum where 1/P (ω) has a minimum. Find the
latter: it’s easier!
(d) Important: Calculate/prove all the following only at resonance, i.e., set
ω = ωo .
Calculate ZL and ZC and show that ZL = −ZC . Use this to find P (ωo ) and phase angle
φ = θZ .
Calculate the amplitudes IL,o ≡ |IL | and IC,o ≡ |IC | of the alternating currents I˜L (t) and
I˜C (t), and show that IL,o = IC,o .
6
Physics 3900
Spring 2016
University of Georgia
Instructor: HBSchüttler
Prove that the L-current exactly cancels the C-current and that the total current equals the
R-current, i.e., at resonance:
˜ = I˜R (t) at all times t
I˜L (t) + I˜C (t) = 0 and I(t)
Hints: First show ZL = −ZC . Then, to calculate current amplitudes and show current
cancellation, use the generalized Ohm’s Law for complex amplitudes I, IL and IC to express
IL and IC in terms of Y , ZL , ZC and I: IL = V /ZL = I/(Y ZL ), likewise IC = V /ZC =
I/(Y ZC ). Hence, |IL | = Io /(|Y ||ZL |) and |IC | = Io /(|Y ||ZC |).
Then use the results for ZL and ZC to show the cancellation in terms of the complex amplitudes: IL = −IC . Hence, |IL | = |IC | and by Kirchoffs junction rule applied to the complex
amplitudes, i.e., by I = IR + IL + IC , show that I = IR .
Multiply the foregoing relations for the complex amplitudes by eiωo t , take the real parts, and
you’ll get the corresponding desired relations for the real alterating currents, I˜L (t), I˜C (t),
˜ and I˜R (t).
I(t)
HWP 02.05: Let the MacLaurin series’ of the complex exponential function, the complex
sine function, and the complex cosine function be defined, for any complex z and integer
n ≥ 0, by
n
X
1 j
z ,
expn (z) :=
j!
j=0
n
X
(−1)j 2j+1
sinn (z) :=
z
,
(2j
+
1)!
j=0
cosn (z) :=
n
X
(−1)j
j=0
(2j)!
z 2j
Use d’Alembert’s Ratio Test to prove that each of these three MacLaurin series has an
infintite radius of convergence, i.e., that each its limit exists for n → ∞ and any complex
number argument z, no matter how large |z|.
Hints: To apply the RT to expn (z), set Tj = (1/j!)z j ; to apply it to sinn (z), set Tj =
[(−1)j /(2j + 1)!]z 2j+1 ; etc. ...
Footnote: The corresponding infinite-series limits can then be used to define the complex
generalizations of the exponential, sine and cosine functions. That is, for any complex
argument z, one defines
∞
X
1 j
z ,
e ≡ exp(z) :=
j!
j=0
z
∞
X
(−1)j 2j+1
sin(z) :=
z
,
(2j + 1)!
j=0
cos(z) :=
∞
X
(−1)j
j=0
(2j)!
z 2j .
For the special case that z ≡ x is a real number, the foregoing definition is consistent with,
i.e., it conincides with, the elementary definitions of the real functions ex , sin x and cos x
because of the fact that the latter three functions can be represented by exactly the same
Mac Laurin series, with z being the real argument x.
7
Physics 3900
Spring 2016
University of Georgia
Instructor: HBSchüttler
HWP 02.06: Find the radius of convergence, Rc , and the center of the convergence disk,
zc , for each of the following four power series, using d’Alembert’s Ratio Test:
∞
X
(z − i + 4)m
,
4m5 + 6m3 + 2
m=0
∞
X
(j + 12)19π 6 −
j=0
∞
z j
X
(−1)j j!
,
i
+
4
j=0
∞
X
j=0
8
(−1)j j!
z 3j
,
5 + 7i
z j!
.
i+4
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